AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH
VOL. 3 NO. 1 (2004)
Apollonius’ Problem: A Study of Solutions and
Their Connections
David Gisch and Jason M. Ribando
Department of Mathematics
University Northern Iowa
Cedar Falls, Iowa 50614-0506 USA
Received: August 15, 2003
Accepted: February 29, 2004
ABSTRACT
In Tangencies Apollonius of Perga showed how to construct a circle that is tangent to three given
circles. More generally, Apollonius' problem asks to construct the circle which is tangent to any
three objects that may be any combination of points, lines, and circles. The case when all three
objects are circles is the most complicated case since up to eight solution circles are possible
depending on the arrangement of the given circles. Within the last two centuries, solutions have
been given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by David
Eppstein in 2001. In this report, we illustrate the solution using the geometry software
Cinderella™, survey some connections among the three solutions, and provide a framework for
further study.
I. INTRODUCTION
Apollonius of Perga was known as
'The Great Geometer'. He should not be
confused with other Greek scholars named
Apollonius, for it was a common name.
Little is known of his life except that he was
born in Perga, Pamphylia, which today is
known as Murtina, or Murtana, and is now in
Antalya, Turkey. The years 262 to 190 B.C.
have been suggested for his life [1-3]. It is
commonly believed that Apollonius went to
Alexandria where he studied under the
followers of Euclid and possibly taught there
later.
This paper focuses on a problem
solved by Apollonius in his book
Tangencies. Apollonius’ works have had a
great influence on the development of
mathematics [4]. In particular, his famous
book Conics introduced terms which are
familiar to us today such as parabola, ellipse
and hyperbola. In Book IV of the Elements,
Euclid details how to construct a circle
tangent to three sides of a given triangle
(Proposition 4) and how to construct a circle
containing three noncollinear points
(Proposition 5) [5, p. 182]. The latter
construction is accomplished by finding the
intersection point of the perpendicular
bisectors of any two sides of the triangle
with the three given points as vertices. In
Tangencies Apollonius poses a
generalization to Euclid’s two propositions:
given any three points, lines or circles in the
plane, construct a circle which contains the
points and is tangent to the lines and circles.
Apollonius enumerated the ten combinations
of points, lines and circles and solved the
cases not already solved by Euclid [6,
p.182]. The case when all three objects are
circles is the most complicated of the ten
cases since up to eight solution circles are
possible depending on the arrangement of
the circles (see Figure 1).
Since no copy of Apollonius’
Tangencies has survived the ages, Pappus
of Alexandria deserves credit for eternally
linking Apollonius’ name with the tangents
problem. Pappus, who lived some five
centuries after Apollonius and is known
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Figure 1. Eight solution circles to Apollonius’ problem.
mostly for his encyclopedic recording and
commenting of Greek mathematics, wrote a
“Treasury of Analysis” in Book VII of his
Mathematical Collections dedicated to works
of Euclid, Aristaeus, and Apollonius. Here
he succinctly states Apollonius’ problem,
acknowledges the ten cases, and provides a
compass and straightedge solution for at
least one solution circle [6, p. 182].
Excepting Arabic reconstructions of
Apollonius’ works, Apollonius’ problem lay
dormant in the literature until François Viète
(1540-1603) restored the Tangencies in
1603. Having unveiled Apollonius’ solution,
Viète challenged Adrianus Romanus to draw
a circle tangent to three given circles but
was disappointed by Romanus’ use of
conics. A century later Newton also went
beyond compass and straightedge solutions
by employing hyperbolas [7]. Agreeing with
Viète’s preference for compass and
straightedge solutions, we are motivated to
include three related constructions under the
same cover: the Euler-Gergonne-Soddy
triangle, which contains the centers of the
two solution circles in the special case when
the circles are mutually tangent or “kissing;”
a solution found by David Eppstein in 2001
for the same special case; and an
adaptation to Gergonne’s analytic solution
which constructs all eight solution circles in
the general case for three circles in generic
position. We conclude with some
connections among the three solutions and
provide a framework for further study
beyond the scope of this paper.
II. THE
EULER-GERGONNE-SODDY
TRIANGLE OF A TRIANGLE
A special case of Apollonius'
problem is known today as the three coins
problem, or kissing coins problem. In this
variant, the three circles, of possibly different
radii, are taken to be mutually tangent.
There are two solutions to this special case
of Apollonius’ problem: a small circle where
all three given circles are externally tangent,
and a large circle where the three given
circles are internally tangent. In 1643 Renè
Descartes sent a letter to Princess Elisabeth
of Bohemia in which he provided a solution
to this special case of Apollonius’ problem.
His solution became known as Descartes’
circle theorem. Philip Beecroft, an English
amateur mathematician, rediscovered
Descartes’ circle theorem in 1842. Then it
was discovered again in 1936 by Frederick
Soddy (1877-1956), who had won a Nobel
Prize in 1921 for his discovery of isotopes
[8]. Soddy expressed the theorem in the
form of a poem, "The Kiss Precise," which
was published in the journal Nature and is
included below. It may have been the flavor
of the added poem that set Soddy apart
from his predecessors, as the two circles are
known today as the inner and outer Soddy
circles. Additionally, Soddy extended the
theorem to the analogous formula for six
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spheres in three dimensions [9, Soddy
Circles].
Though Soddy provided an analytic
solution to the kissing coins problem, we do
not know whether he constructed a synthetic
solution using compass and straightedge.
For our part we find compass and
straightedge solutions easier to understand
than their analytic counterparts. This
section is devoted to the Euler-Gergonne-
Soddy triangle (EGST). Though it does not
provide a solution to the kissing coins
problem, it does provide a unique insight to
finding the Soddy circles. Many of the
connections between the EGST and the
Soddy circles solution to the kissing coins
problem were inspired by Oldknow’s article
[8], which uses trilinear coordinates,
harmonic ranges, and parameterizations of
lines to construct the EGST. It is interesting
to note that Oldknow attributes many of his
investigations to the use of geometric
software packages, a current trend among
geometry researchers.
As noted already, the kissing coins
problem is the special case of Apollonius’
problem where all three circles are mutually
tangent. In this arrangement there are two
solution circles, the inner and outer Soddy
circles as shown in Figure 2. There are two
questions of interest that arise from this
special case: what are the radii of the Soddy
circles and where are their centers located?
As a partial answer to the latter
question, the Soddy centers lie on the line
Figure 2. Soddy circles.
The Kiss Precise
For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb,
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
*The sum of the squares of all four bends
Is half the square of their sum.*
To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
*The square of the sum of all five bends
Is thrice the sum of their squares.*
Frederick Soddy (Nobel Prize, Chemistry,
1921) Nature, June 20, 1936 [10].
determined by the incenter and Gergonne
point of the reference triangle. This line is
known as the Soddy line and it contains one
of the sides of the EGST. Given three
vertices of a triangle, we show how to
construct its EGST. The reader may want to
refer to the Appendix for a glossary of
geometric terms involved in this
construction.
We start with the Euler line since it
is the easiest to construct. Given a
reference triangle, find the circumcenter as
the intersection of the perpendicular
bisectors and the orthocenter as the
intersection of the altitudes. These two
points determine the Euler line, which also
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Figure 3. The Gergonne point, incenter, incircle and Soddy line.
contains the triangle’s centroid (the
intersection of the medians). The Euler line
also contains a number of other important
triangle centers including the center of the
nine-point circle.
The Soddy line is determined by the
incenter, where the angle bisectors coincide,
and the Gergonne point. The incenter is the
center of the unique circle that is internally
tangent to its reference triangle at three
points. These three points are called the
contact points, and together they form the
contact triangle of the reference triangle.
The Gergonne point can most easily be
found as follows: erect perpendiculars
containing the incenter from each side of the
reference triangle. These perpendiculars
meet the reference triangle at the contact
points. The lines containing the contact
points and the opposite vertices on the
reference triangle coincide at the Gergonne
point (see Figure 3).
Importantly, the contact points have
special meaning to the kissing coins
problem. Given three noncollinear points,
one can uniquely construct the three
mutually tangent circles centered at these
points. Construct the reference triangle with
these vertices and its contact triangle, as
described above; the mutually tangent
circles are centered at the vertices of the
reference triangle and contain the nearby
vertices of the contact triangle! Hence,
three non-collinear points uniquely
determine the radii of the kissing coins.
To construct the Gergonne line, one
must understand the notion of triangles in
perspective. The idea of perspective was
introduced in the Renaissance to create the
idea of depth in art. The principle of
perspective is that all lines meet at a point,
thus providing depth. A well-known example
of this is Leonardo da Vinci’s Last Supper,
where all the lines forming the walls, ceiling
and edges of the table meet at a point of
perspective, which happens to be the head
of Christ.
The idea of perspective objects can
be applied in geometry in relation to
triangles. The two triangles ∆ABC and
∆A'B'C' in Figure 4 are perspective from a
line since the extensions of their three pairs
of corresponding sides meet in collinear
points X, Y, and Z. The line joining these
points is called the perspectrix. It can also
be said that two triangles are perspective
from a point if their three pairs of lines
joining their corresponding sides meet at a
point of concurrence O. This point is called
the
perspector,
perspective center,
homology center, or pole [9, Perspective
Triangles].
In the kissing coins problem the
triangle formed by the centers of the circles
and its contact triangle are perspective
triangles. The perspector and the
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Figure 4. Triangles in perspective: Triangles ∆ ABC and ∆ A'B'C' are perspective from a line since
the extensions of their three pairs of corresponding sides meet in collinear points X, Y, and Z.
perspectrix of these two triangles are called
the Gergonne point and Gergonne line,
respectively, as shown in Figure 5.
The union of the Euler line, Soddy
line and Gergonne line form the EGST, as
seen in Figure 6. The vertices of EGST are
known as the de Longchamps point, Evans
point and Fletcher point. The Soddy line
and Gergonne line always form a right angle
at the Fletcher point [8, p. 328]. As
mentioned previously the Soddy points,
being the centers of the inner and outer
Soddy circles, lie on the Soddy line and form
a harmonic range with the incenter and
Gergonne point [8, p. 326]. Further, the
radii of both Soddy circles can be expressed
in terms of ratios of the radii of the three
given circles and the incircle. We will see in
the next section that two of the solution
circles to Apollonius’ problem are centered
on the Soddy line of the EGST.
Figure 5. Gergonne point and line.
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Figure 6. Euler-Gergonne-Soddy triangle.
III.
THE EPPSTEIN SOLUTION
David Eppstein, a professor in the
Department of Information and Computer
Science at the University of California at
Irvine, is a researcher in the areas of
computational geometry and graph
algorithms. Eppstein is also the founder and
author of the popular Geometry Junkyard
website. On this site Eppstein details a
solution, discovered by Eppstein himself,
that constructs Frederick Soddy’s circles
with a compass and straightedge. The basis
of Eppstein’s solution comes from an article
in The American Mathematical Monthly [13].
In this article Eppstein proves that three
lines through opposite points of tangency of
any four mutually tangent spheres in three-
space are coincident. A resulting corollary
from this lemma is that three lines through
opposite points of tangency of any four
mutually tangent circles in the plane are
coincident.
Eppstein’s solution is as follows
(see Figure 7a): form a triangle connecting
the three circle centers and drop a
perpendicular line from each center to the
opposite triangle edge. Each of these lines
cuts its circle at two points. Seen in Figure
7b, construct a line from each cut point to
the point of tangency of the other two
circles. These lines cut their circles in two
more points, yielding six total, which are the
points of tangency of the Soddy circles.
Once these six points are known, the Soddy
circles’ centers are easily found to lie on the
line determined by the incenter and
Gergonne point, a.k.a. the Soddy line.
In Eppstein’s solution, one may
notice that there are two sets of three lines,
as seen in Figure 7b, each intersecting at a
common point. This is the result of the
corollary from Eppstein’s article mentioned
above. Eppstein points out that despite their
simplicity of definition and the large amount
of study into triangle geometry, these two
points do not appear in the list of over 1,000
known triangle centers collected by Clark
Kimberling and Peter Yff [13, p. 65]. Thus,
these two points have become known as the
Eppstein points and, remarkably, they lie on
the Soddy line and form a harmonic range
with the Gergonne point and incenter [8, p.
327].
Figure 7a. The Eppstein solution.
Figure 7b. Soddy circle and Eppstein
points.
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Figure 8. Inversion of line l through circle C.
IV.
THE GERGONNE SOLUTION
Joseph Diaz Gergonne was born in
France in 1771 and died there in 1859. He
spent most of his youth serving in the
military until 1795. Afterwards, he began his
mathematical study, which spawned a
number of mathematical ideas but mostly
focused on the area of geometry. Gergonne
is well known for creating the journal
officially called the Annales de
Mathématique Pures et Appliquées but
became known as Annales de Gergonne
[14, p. 226]. His journal featured such
prominent mathematicians as Jakob Steiner
and Evariste Galois. Mentioned earlier are a
few of the results of his work in the solution
of the Soddy circles. Not only did Gergonne
supply all eight solution circles to Apollonius’
problem, he also introduced the word polar
as it applies to inversion geometry.
Inversion geometry deals with
transformations of the plane that leave a
given circle fixed while taking its interior
points to its exterior and vice versa. Not just
the subject of advanced Euclidean
geometry, inversion geometry arises in
hyperbolic geometry and conformal
mappings of the complex plane. While the
details and intricacies of inversion geometry
are beyond the scope of this paper, we do
make use of the fact that we can invert any
point through a given circle using compass
and straightedge constructions [15].
For our purposes, we need the
inversion geometry fact that a circle
inversion through the circle C of radius r
centered O in Figure 8 takes the line l to the
circle with diameter
OQ
. In this
arrangement P and Q are on the line
perpendicular to l containing O, and
(OP)(OQ) = r
2
. Points P and Q are easily
constructed with compass and straightedge,
and point Q is called the inversion pole of
the line l.
Gergonne’s solution requires
constructing the dilation points for each pair
of circles [16]. The dilation points of a pair
of circles are the two points of central
similarity about which one circle can be
dilated (or contracted) to the other. As there
are three pairs of circles with two dilation
points each, this process yields six points.
These lie three by three on four lines,
forming a four-line geometry, as illustrated in
Figure 9.
Determine the inversion poles of
one of the dilation lines with respect to each
of the three circles, as in Figure 10a, and
connect the inversion poles with the radical
center, as shown in Figure 10b. The radical
center is the intersection of the three radical
axes; the radical axis of two circles is the
line that contains the center points of all
Figure 9. Dilation points and lines
Figure 10a. Inversion poles
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Figure 10b. Tangent points.
Figure 11a. Tangent circle.
Figure 11b. Another tangent circle.
that are orthogonal to both of the given
circles. In this case, each line containing the
radical center and an inversion pole
intersects its respective circle at two points.
The center of the upper circle in
Figure 10b is alone on one side of the
dilation line formed by the dilation points. On
that circle pick the intersection point furthest
from the radical center. On the other two
circles pick the near intersection points.
These three points are the tangent points for
the solution circle shown in Figure 11a. The
other three intersection points are the
tangent points for another solution circle,
shown in Figure 11b.
This construction sequence yields
all four lines formed from the dilation points,
and for each line produces at most two
solution circles. Thus, all eight solution
circles can be constructed. We should
mention that this process, while not terribly
complicated, does require careful record
keeping. When constructing Gergonne’s
solution, we used the interactive geometry
software program Cinderella™.
Cinderella™ allows one to hide lines and
circles. Without this capability we do not
believe that we could have constructed the
solution. As pictured below in Figure 12, the
solution becomes very cluttered when every
line and circle is visible. We are truly
impressed that Gergonne, or anyone before
the computer age, could have the patience
to perform this feat.
V. CONNECTIONS
AND
EXTENSIONS
In this section we observe some
connections relating the EGST, Eppstein’s
solution and Gergonne’s solution and
suggest some routes for further study.
One of the most intriguing properties
that we have found is the relation of the
Gergonne line and the Gergonne solution.
Since both share the name of J. D.
Gergonne it might come as no surprise that
they are related, but we do not consider this
fact to be obvious. In Gergonne’s solution
one may recall that a line formed by the
three outer dilation points, which we call the
outer dilation line, is used to find two
solution circles. In the special case where
all three circles are mutually tangent, the
Gergonne line and the outer dilation line
coincide. Also, when the given circles are
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Figure 12. The Complete Gergonne Solution.
mutually tangent, the two solution circles
formed by the outer dilation line are the
Soddy circles. Thus, one begins to see the
importance of the Euler-Gergonne-Soddy
triangle and the role it plays in Apollonius’
problem. Furthermore, if the three given
circles are mutually tangent, the incenter of
the reference triangle coincides with the
radical center, and the radical axes that
define the radical center are the
perpendicular lines from the triangle sides to
the incenter.
Until now we have allowed the
reader to believe that Eppstein’s solution to
the kissing coins problem only solves for two
of the solution circles whereas Gergonne’s
solution finds all eight solution circles to the
general Apollonius’ problem. The observant
reader should ask: “Where are the
remaining six tangent circles in the kissing
coins arrangement?” It turns out that each
of the three given circles represents two
solutions. This can best be observed using
geometry software to execute Gergonne’s
solution and arranging the three given
circles to be nearly tangent. One will
observe that the coins, now separated by a
small distance from each other, are each
internally tangent to two solution circles just
slightly larger than the coin itself.
As another special case of three
circles in general position, consider what
happens when the radius of one of the given
circles tends toward zero. Here, the eight
solution circles collapse to four, and they do
so in pairs. When the radius of a second
circle decreases, the four solution circles
collapse to two. Finally, when the radius of
the third circle shrinks, the two solutions
collapse to one. Figure 13 illustrates this
nicely for a sequence of circles whose radii
tend toward zero. Hence, Euclid’s
Proposition 5, Book IV, for finding the
circumcenter of a triangle should really be
viewed as a very special case of Gergonne’s
solution to Apollonius’ problem!
As yet another special case,
consider when the given circles are the
excircles of a triangle. Here, the vertices of
the triangle are the inner dilation points for
the pairs of circles. Surprisingly, the lines
extending the triangle sides are three of the
Apollonius solutions; they are limiting cases
as the radii of three of the solution circles
tend to infinity! Feuerbach’s Theorem
guarantees that the nine-point circle is
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Figure 13. Shrinking circles and collapsing solutions.
simultaneously tangent to the three
excircles, providing a fourth solution circle
[17,p. 46 ]. This solution is analogous to the
inner Soddy circle solution to the kissing
coins problem in that the three excircles are
externally tangent to the nine-point circle. A
fifth solution circle lies so that the excircles
are internally tangent, and the remaining
three solution circles each have one excircle
internally tangent and the other two excircles
externally tangent.
With regard to further study, the
interested reader may wish to pursue the
following topics.
In Apollonius’ problem the three
given objects are taken from among circles,
points and lines. The latter two objects
should be thought of as limiting cases when
the radius of a circle approaches zero or
infinity. Gergonne’s solution only applies to
the case when all three objects are circles
since circle inversion is used. Rather than
use three circles, one may wish to examine
all combinations of points, lines and circles,
decide for which configurations all eight
solution circles appear, and use geometry
software to construct solution circles. For
example, no solution circles exist in the case
of three parallel lines, and two solution
circles exist for two parallel lines cut by a
transversal. Are there any connections to
Gergonne’s solution?
It is also worth exploring the special
case of three circles centered at points that
are the vertices of an isosceles or an
equilateral triangle. If at least two of the
three sides of the reference triangle are
congruent, the Euler line and the Soddy line
coincide, so the EGST is degenerate. In the
context of Eppstein’s solution, this
corresponds to kissing coins with equal radii.
Given three noncollinear points,
consider all sets of three non-overlapping
circles centered at these points. What are
the possible locations for the centers of the
tangency circles for which the three circles
are either all internally tangent or all
externally tangent? The kissing coins
problem is the special case when the three
circles centered at the triangle vertices are
tangent to each other. In this instance,
perhaps the inner and outer Soddy centers
that solve this problem are known triangle
centers.
Finally, as noted in section 2, the EGST
is always a right triangle. A benefit of
Cinderella™ is the ability to dynamically
move elements while preserving the incident
relations among points, lines and circles.
With this capability the vertices of the
original triangle can be moved freely, and
we have observed that the EGST always
appears to be long and skinny (see Figure
6). The maximum value of the angle
between the Euler line and Gergonne line is
90 degrees, but what is the smallest
possible value of this angle?
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APPENDIX: Glossary of Terms
Points and Centers
Centroid
The intersection of the medians
Circumcenter The
intersection
of the perpendicular bisectors
Incenter
Intersection of the angle bisectors
Orthocenter
Intersection of the altitudes
Gergonne point
Perspector of a reference triangle and its contact triangle
Evans point
Intersection of the Euler and Soddy lines
Fletcher point
Intersection of the Soddy and Gergonne lines
De Longchamps
Intersection of the Euler and Gergonne lines
Lines
Euler line
Defined by the circumcenter and orthocenter; also contains the
centroid and nine-point center
Gergonne line
Line of perspective for a reference triangle and its contact triangle
Soddy line
Defined by the incenter and Gergonne point; also contains the
Eppstein and Soddy points
Circles
Circumcircle
Unique circle containing a triangle’s vertices
Incircle
Unique circle internally tangent to all three triangle sides
Nine-point circle
Contains side midpoints, feet of altitudes, and midpoints of
segments joining the orthocenter to the vertices
Inner Soddy circle
Circle that is internally tangent to the kissing coins
Outer Soddy circle
Circle that is externally tangent to the kissing coins
Triangles
Contact triangle
Defined by the points of tangency of reference triangle and its
incircle; vertices are where the kissing coins touch
Euler-Gergonne-Soddy
triangle
Triangle formed by the Euler, Gergonne and Soddy lines
25
AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH
VOL. 3 NO. 1 (2004)
26
Document Outline - Apollonius’ Problem: A Study of Solutions and The
-
- David Gisch and Jason M. Ribando
- The Kiss Precise
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